Polynomial

A function f(x) defined by f(x) = a0 xn + a1xn-1 +……..an-1x + an (a0 ¹ 0) where n is non-negative integer and a0, a1, a2…,an-1, an are constants is called polynomial of degree n in x.

In particulars, the functions

f(x) = ax + b, g(x) = ax2 + bx + c, h(x) = ax3 + bx2 + cx + d respectively are the linear, quadratic and cubic polynomial.

Polynomial equation

The equation of the form

a0xn + a1xn-1 + …….+ an-1x + an = 0 (a ¹), where n is non-negative integer and a0, a1, a2 ….. an are constants is called polynomial equation of degree n in x.

In particular, ax + b = 0, ax2 + bx + c = 0 and ax3 + bx3 + cx + d = 0 respectively are the linear, quadratic and cubic equations.

Quadratic equation

The equation of the form ax2 + bx + c = 0 (a ¹ 0) is called quadratic equation.

The roots of the quadratic equation are given by 


Nature of the roots of quadratic equation

·         If b2 - 4ac > 0, then the roots of ax2 + bx + c = 0 are real and unequal. In particular if b2 -     4ac > 0 and perfect square, then the roots are rational and unequal.

·         If b2 - 4ac = 0, then the roots are real and equal, each being b/2a.

·         If b2 - 4ac < 0, then the roots of quadratic equation are imaginary and unequal.

·       The irrational and imaginary roots of quadratic equation always occur in pair and each being     the conjugate of the other.

                    Relation between the roots and the coefficients